Grid-Based Graphs, Linear Realizations and the Buratti-Horak-Rosa Conjecture

Label the vertices of the complete graph Kᵥ with the integers \0, 1, , v-1\ and define the length of the edge between distinct vertices labeled x and y by (x, y) = (|y-x|, v - |y-x|). A realization of a multiset L of size v-1 is a Hamiltonian path through Kᵥ whose edge labels are L. The Buratti-Horak-Rosa (BHR) Conjecture is that there is a realization for a multiset L if and only if for any divisor d of v the number of multiples of d in L is at most v-d. We introduce ``grid-based graphs" as a useful tool for constructing particular types of realizations, called ``linear realizations, " especially when the multiset in question has a support of size 3. This lets us prove many new instances of the BHR Conjecture, including those for multisets of the form \1ᵃ, xᵇ, yᶜ \ when a x+y -, where is the number of even elements in \ x, y \, and those for all multisets of the following forms for sufficiently large v with (v, y) = 1 for all y L: \1ᵃ, 2ᵇ, xᶜ\, except possibly when a \1, 2\ and x is odd, \1ᵃ, xᵇ, (x+1) ᶜ\. This establishes that there are infinitely many sets U of size 3 for which there are infinitely many values of v where the BHR Conjecture holds for each multiset with support U. We also show that the BHR Conjecture holds for \1ᵃ, xᵇ, (x+1) ᶜ\ when x \7, 9, 10\ and (v, x) = (v, x+1) = 1.